Math.Content.HSF-IF: Interpreting Functions

Math.Content.HSF-IF.A: Understand the concept of a function and use function notation

Math.Content.HSF-IF.A.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Introduction to Functions
Linear Functions
Points, Lines, and Equations

Math.Content.HSF-IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Absolute Value with Linear Functions
Exponential Functions
Points, Lines, and Equations
Quadratics in Polynomial Form

Math.Content.HSF-IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Arithmetic Sequences
Geometric Sequences

Math.Content.HSF-IF.B: Interpret functions that arise in applications in terms of the context

Math.Content.HSF-IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Absolute Value with Linear Functions
Exponential Functions
General Form of a Rational Function
Graphs of Polynomial Functions
Logarithmic Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Slope-Intercept Form of a Line
Standard Form of a Line

Math.Content.HSF-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Absolute Value with Linear Functions
Exponential Growth and Decay
Points, Lines, and Equations

Math.Content.HSF-IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Distance-Time Graphs
Distance-Time Graphs - Metric
Distance-Time and Velocity-Time Graphs
Distance-Time and Velocity-Time Graphs - Metric
Slope

Math.Content.HSF-IF.C: Analyze functions using different representations

Math.Content.HSF-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Math.Content.HSF-IF.C.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Point-Slope Form of a Line
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic
Slope-Intercept Form of a Line
Standard Form of a Line
Translating and Scaling Functions

Math.Content.HSF-IF.C.7.b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Absolute Value with Linear Functions
Radical Functions
Translating and Scaling Functions

Math.Content.HSF-IF.C.7.c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graphs of Polynomial Functions
Polynomials and Linear Factors

Math.Content.HSF-IF.C.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

General Form of a Rational Function
Rational Functions

Math.Content.HSF-IF.C.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Exponential Functions
Introduction to Exponential Functions
Logarithmic Functions
Logarithmic Functions: Translating and Scaling
Translating and Scaling Functions

Math.Content.HSF-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Math.Content.HSF-IF.C.8.a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Quadratics in Factored Form

Math.Content.HSF-IF.C.8.b: Use the properties of exponents to interpret expressions for exponential functions.

Compound Interest
Exponential Growth and Decay

Math.Content.HSF-BF: Building Functions

Math.Content.HSF-BF.A: Build a function that models a relationship between two quantities

Math.Content.HSF-BF.A.1: Write a function that describes a relationship between two quantities.

Math.Content.HSF-BF.A.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

Math.Content.HSF-BF.A.1.b: Combine standard function types using arithmetic operations.

Addition and Subtraction of Functions

Math.Content.HSF-BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Arithmetic Sequences
Arithmetic and Geometric Sequences
Geometric Sequences

Math.Content.HSF-BF.B: Build new functions from existing functions

Math.Content.HSF-BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Exponential Functions
Graphs of Polynomial Functions
Introduction to Exponential Functions
Logarithmic Functions: Translating and Scaling
Quadratics in Polynomial Form
Quadratics in Vertex Form
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Zap It! Game

Math.Content.HSF-BF.B.4: Find inverse functions.

Math.Content.HSF-BF.B.4.a: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

Logarithmic Functions
Radical Functions

Math.Content.HSF-BF.B.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

Logarithmic Functions

Math.Content.HSF-BF.B.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Logarithmic Functions

Math.Content.HSF-LE: Linear, Quadratic, and Exponential Models

Math.Content.HSF-LE.A: Construct and compare linear, quadratic, and exponential models and solve problems

Math.Content.HSF-LE.A.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.

Math.Content.HSF-LE.A.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Arithmetic and Geometric Sequences

Math.Content.HSF-LE.A.1.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Arithmetic Sequences
Arithmetic and Geometric Sequences

Math.Content.HSF-LE.A.1.c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Arithmetic and Geometric Sequences
Compound Interest
Exponential Growth and Decay
Geometric Sequences

Math.Content.HSF-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Arithmetic Sequences
Arithmetic and Geometric Sequences
Compound Interest
Exponential Functions
Exponential Growth and Decay
Geometric Sequences
Introduction to Exponential Functions
Point-Slope Form of a Line
Slope-Intercept Form of a Line
Standard Form of a Line

Math.Content.HSF-LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Arithmetic and Geometric Sequences

Math.Content.HSF-LE.A.4: For exponential models, express as a logarithm the solution to (ab)^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Logarithmic Functions

Math.Content.HSF-LE.B: Interpret expressions for functions in terms of the situation they model

Math.Content.HSF-LE.B.5: Interpret the parameters in a linear or exponential function in terms of a context.

Compound Interest
Exponential Growth and Decay
Slope-Intercept Form of a Line
Standard Form of a Line

Math.Content.HSF-TF: Trigonometric Functions

Math.Content.HSF-TF.A: Extend the domain of trigonometric functions using the unit circle

Math.Content.HSF-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Cosine Function
Radians
Sine Function
Tangent Function

Math.Content.HSF-TF.A.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Cosine Function
Sine Function
Tangent Function

Math.Content.HSF-TF.A.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi – x, pi + x, and 2pi – x in terms of their values for x, where x is any real number.

Cosine Function
Sine Function
Tangent Function

Math.Content.HSF-TF.A.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function
Sine Function
Tangent Function

Math.Content.HSF-TF.B: Model periodic phenomena with trigonometric functions

Math.Content.HSF-TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Sine Function
Sound Beats and Sine Waves
Translating and Scaling Sine and Cosine Functions
Waves

Math.Content.HSF-TF.C: Prove and apply trigonometric identities

Math.Content.HSF-TF.C.8: Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.

Cosine Function
Sine Function

Math.Content.HSF-TF.C.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Sum and Difference Identities for Sine and Cosine